Fixed-Parameter Tractability and Completeness II:On Completeness for W[1]

Rod G. DowneyMathematics Department

Victoria UniversityWellington, New Zealand

Michael R. FellowsComputer Science Department

University of VictoriaVictoria, B.C. Canada

Abstract

For many fixed-parameter problems that are trivially solvable in polynomial-time, such as k-

DOMINATING SET, essentially no better algorithm is presently known than the one which tries all

possible solutions. Other problems, such as FEEDBACK VERTEX SET, exhibit fixed-parameter

tractability: for each fixed k the problem is solvable in time bounded by a polynomial of degree c,

where c is a constant independent of k. In a previous paper, the W Hierarchy of parameterized

problems was defined, and complete problems were identified for the classes W [t] for t ≥ 2. Our

main result shows that INDEPENDENT SET is complete for W [1].

1. Introduction

Many natural computational problems have input that consists of a pair of items. Forpractical applications, it is often the case that only a small range of parameter values aresignificant.

We now have encouraging fixed-parameter tractability results for many problems. Forexample, for each fixed parameter value k, it can be determined whether a graph G on nvertices has k disjoint cycles in time O(n) [Bo,DF1]. MINOR TESTING and the GRAPHGENUS problem can be solved in O(n3) time for each fixed parameter value by the deepresults of Robertson and Seymour [RS1,RS2].

There are many other parameterized problems, such as DOMINATING SET, for whichessentially no better algorithm is presently known than the trivial brute-force algorithm thatchecks all sets of k vertices.

The following definitions provide a framework for the study of fixed-parameter complex-ity.

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Definition. A parameterized problem is a set L ⊆ Σ∗ × Σ∗ where Σ is a fixed alphabet.

Definition. A parameterized problem L is (uniformly) fixed-parameter tractable if thereexists a constant α and an algorithm to determine if (x, y) is in L in time f(|y|) · |x|α, wheref : N → N is an arbitrary function. We will denote the class of fixed-parameter tractableproblems by FPT .

In a previous paper we defined a hierarchy of parameterized problem classes

FPT ⊆ W [1] ⊆ W [2] ⊆ W [3] ⊆ · · · ⊆ W [SAT ] ⊆ W [P ]

and exhibited problems complete for W [t] for t ≥ 2. For example, DOMINATING SET iscomplete for W [2]. Our main result in the present paper shows that several natural problems(including INDEPENDENT SET) are complete or hard for W [1]. We remark that W [1] iscurrently the most important of the parameterized classes that we beleive to be intractable.This is because it is our current “minimally intractable” class in the sense that we believeit to be intractable and if we wish to prove a problem to be fixed parameter intractable wewill establish this by showing hardness for W [1]. The reasons that we believe that W [1] isfixed parameter intractable are that many problems have been shown to be W [1] completehere and elsewhere (e.g. [CCDF]) and the following generic problem is W [1] complete (in[CCDF]):

SHORT TURING MACHINE COMPUTATION

Input: A Nondeterministic Turing Machine M and a string x.Parameter: kQuestion: Does M have a computation path that accepts x in at most k steps?

We believe that the W [1] completeness of this problem establishes a miniaturized Cook-Levin theorem and provides very strong evidence that W [1] really is fixed parameter in-tractable.

For a parameterized problem L and y ∈ Σ∗ we write Ly to denote the associated fixed-parameter problem (y is the parameter) Ly = {x|(x, y) ∈ L}.

Definition. A (uniform) reduction of a parameterized problem L to a parameterized problemL′ is an oracle algorithm A that on input (x, y) determines whether x ∈ Ly and satisfies(1) There is an arbitrary function f : N → N and a polynomial q such that the runningtime of A is bounded by f(|y|)q(|x|).(2) For each y ∈ Σ∗ there is a finite subset Jy ⊆ Σ∗ such that A consults oracles only forfixed-parameter decision problems L′w where w ∈ Jy.

If, additionally the functions f and y → Jy are both recursive we say that the reductionis strongly uniform. (All of the reductions in this paper are strongly uniform.)

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A motivating example for the above definition is the reduction of the Graph Genusproblem to the problem of MINOR TESTING. By the deep results of Robertson and Seymour[RS1,RS2] the GRAPH GENUS problem for each fixed parameter value k reduces to finitelymany minor tests; the reduction can be made uniform by the techniques of [FL1,FL2]. Thefollowing is easily verified.

Lemma 1.1 If the parameterized problem L reduces to the parameterized problem L′, andif L′ is f.p. tractable, then L is f.p. tractable. 2

In the Section 2 we review the definition of the W hierarchy. In Section 3 we proveour main result, that INDEPENDENT SET is complete for W [1]. In Section 4 we discussa number of natural problems that are hard for W [1], including a parameterized variant ofSUBSET SUM. Section 5 concludes with a discussion of open problems. We remark thatthe results of this paper have been used in many W [1] hardness proofs, as well as appliedto Computational Learning Theory([DEF]). Moreover, as we montioned earlier, since thewriting of the present paper, many other W [1] hardness and completeness results have beenfound. We make some further remarks towards this in an Addendum in §6.

2. The W Hierarchy of Parameterized Problems

The complexity classes W [t] of parameterized problems intuitively reflect the difficultyof checking a solution. We first define circuits in which some gates have bounded fan-in andsome have unrestricted fan-in. It is assumed that fan-out is never restricted.

Definition. A Boolean circuit is of mixed type if it consists of circuits having gates of thefollowing kinds.

(1) Small gates: not gates, and gates and or gates with bounded fan-in. We will usuallyassume that the bound on fan-in is 2 for and gates and or gates, and 1 for not gates.

(3) Large gates: And gates and Or gates with unrestricted fan-in.

We will use lower case to denote small gates (or gates and and gates), and upper caseto denote large gates (Or gates and And gates).

Definition. The depth of a circuit C is defined to be the maximum number of gates (smallor large), not counting not gates, on an input-output path in C. The weft of a circuit C isthe maximum number of large gates on an input-output path in C.

Definition. We say that a family of circuits F has bounded depth if there is a constant h suchthat every circuit in the family F has depth at most h. We say that F has bounded weft ifthere is constant t such that every circuit in the family F has weft at most t. F is a decisioncircuit family if each circuit has a single output. A decision circuit C accepts an input vectorx if the single output gate has value 1 on input x. The weight of a boolean vector x is the

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number of 1’s in the vector.

Definition. Let F be a family of decision circuits. We allow that F may have many differentcircuits with a given number of inputs. To F we associate the parameterized circuit problemLF = {(C, k) : C ∈ F and C accepts an input vector of weight k}.

Definition. A parameterized problem L belongs to W [t] (monotone W [t]) if L uniformlyreduces to the parameterized circuit problem LF (t,h) for the family F (t, h) of mixed type(monotone) decision circuits of weft at most t, and depth at most h, for some constant h.

Thus we have the containments

FPT ⊆ W [1] ⊆ W [2] ⊆ ...

and we conjecture that each of these containments is proper. We term the union of theseclasses the W hierarchy. If we place no bound on the depth or weft of the circuits we similarlyget the class W .

By definition, a parameterized problem L ∈ W [1] reduces to LF (1,h) for the family F (1, h)of weft 1 circuits of depth bounded by h, for some h. The following argument shows that wemay assume the circuits in F to have depth 2 and a particularly simple form, consisting of asingle output And gate which receives arguments from or gates having fan-in bounded by aconstant h′. Thus each such circuit is isomorphically represented by a boolean expression inconjunctive normal form having clauses with at most h′ literals. We will say that a family ofcircuits having this form is normalized. With this in mind we have the following definition.

Definition. The family of parameterized problems W [1, s] is defined to be those parameter-ized problems in W [1] reducible to LF (s) for the family F (s) of depth 2, weft 1 normalizedcircuits, with the or gates on level 1 having fan-in bounded by s.

Lemma 2.1. Let F be a family of weft 1 circuits of depth bounded by a constant h. ThenLF is reducible to LF (s) for s = 2h + 1, and hence LF ∈ W [1, s].

Proof. Let C ∈ F and let k be a positive integer. We describe how to produce a circuitC ′ ∈ F (s) and an integer k′ such that C accepts a weight k input if and only if C ′ acceptsan input of weight k′.

Step 1. The reduction to tree circuits.

The first step is to transform C into an equivalent weft 1 tree circuit C ′ of depth atmost h. In a tree circuit every logic gate has fan-out one, and thus the circuit can be viewedas equivalent to a Boolean formula. The transformation can be accomplished by replicatingthe portion of the circuit above a gate as many times as the fan-out of the gate, beginningwith the top level of logic gates, and proceeding downward level by level. The creation of C ′

from C may require time O(|C|O(h)) and involve a similar blow-up in the size of the circuit.

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This is permitted since h is a fixed constant independent of k and |C|.

Step 2. Moving the not gates to the top of the circuit.

Let C denote the circuit we receive from the previous step (we will use this notationalconvention throughout the proof). Transform C into an equivalent circuit C ′ by commutingthe not gates to the top, using DeMorgan’s laws. This may increase the size of the circuitby at most a constant factor. The tree circuit C ′ thus consists (from the top) of the inputnodes, with not gates on some of the lines fanning out from the inputs. In counting levelswe consider all of this as level 0.

Step 3. A preliminary depth 4 normalization.

The goal of this step is to produce a tree circuit C ′ of depth 4 that corresponds to aBoolean expression E in the following form. (For convenience we use product notation todenote logical ∧ and sum notation to denote logical ∨.)

E =m∏i=1

mi∑j=1

Eij

where:(1) m is bounded by a function of h(2) for all i, mi is bounded by a function of h(3) for all i, j, Eij is either:

Eij =mij∏k=1

mijk∑l=1

x[i, j, k, l]

or

Eij =mij∑k=1

mijk∏l=1

x[i, j, k, l]

where the x[i, j, k, l] are literals (i.e., input Boolean variables or their negations) and for alli, j, k, mijk is bounded by a function of h. The family of circuits corresponding to theseexpressions has weft 1, with the large gates corresponding to the Eij. (In particular, the mij

are not bounded by a function of h.)

To achieve this form, let g denote a large gate in C. An input to g is computed by asubcircuit of depth bounded by h consisting only of small gates, and so is a function of atmost 2h literals. This subcircuit can thus be replaced, at constant cost, by either a product-of-sums expression (if g is a large ∧ gate), or a sum-of-products expression (if g is a large∨ gate). In the first case, the product of these replacements over all inputs to g yields thesubexpression Eij corresponding to g. In the second case, the sum of these replacementsyields the corresponding Eij.

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The output of C is a function of the outputs of at most 2h large gates. This functioncan be expressed as a product-of-sums expression of size at most 22h . At the cost of possiblyduplicating some of the large gate subcircuits at most 22h times, we can achieve the desirednormal form with the bounds: m ≤ 22h , mi ≤ 2h and mijk ≤ 2h.

Step 4. Employing additional nondeterminism.

Let C denote the normalized depth 4 circuit received from Step 3 and correspondingto the Boolean expression E described above. For convenience, assume that the Eij forj = 1, ...,m′i are sums-of-products and the Eij for j = m′i + 1, ...,mi are products-of-sums.Let V0 = {x1, ..., xn} denote the variables of E.

In this step we produce an expression E ′ in product-of-sums form with the size of thesums bounded by 2h + 1 that has a satisfying truth assignment of weight

k′ = 2k + k(1 + 2h)22h +m+m∑i=1

m′i

if and only if C has a satisfying truth assignment of weight k. The main idea is to useadditional (bounded weight) nondeterminism to guess both: (1) a weight k input x for C,and (2) additional information that will allow us to check that C(x) = 1 with a W [1, s]circuit, s = 2h + 1.

The set V of variables of E ′ is V = V0 ∪ V1 ∪ V2 ∪ V3 where

V1 = {x[i, j] : 1 ≤ i ≤ n, 0 ≤ j ≤ (1 + 2h)22h}

V2 = {u[i, j] : 1 ≤ i ≤ m, 1 ≤ j ≤ mi}

V3 = {w[i, j, k] : 1 ≤ i ≤ m, 1 ≤ j ≤ m′i, 0 ≤ k ≤ mij}

The expression E ′ is a product of subexpressions E ′ = E1 ∧ · · · ∧ E8 described:

E1 =n∏i=1

(¬xi + x[i, 0])(¬x[i, 0] + xi)

E2 =n∏i=1

22h+1∏j=0

(¬x[i, j] + x[i, j + 1 (mod r)]) r = 1 + (1 + 2h)22h

E3 =m∏i=1

mi∑j=1

u[i, j]

E4 =m∏i=1

mi−1∏j=1

mi∏j′=j+1

(¬u[i, j] + ¬u[i, j′])

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E5 =m∏i=1

m′i∏j=1

mij−1∏k=0

mij∏k′=k+1

(¬w[i, j, k] + ¬w[i, j, k′])

E6 =m∏i=1

m′i∏j=1

(¬u[i, j] + ¬w[i, j, 0])

E7 =m∏i=1

m′i∏j=1

mij∏k=1

mijk∏l=1

(¬w[i, j, k] + x[i, j, k, l])

E8 =m∏i=1

mi∏j=m′i+1

mij∏k=1

(¬u[i, j] +

mijk∑l=1

x[i, j, k, l]

)

To see that Step 4 works correctly, suppose τ is a weight k truth assignment to V0 thatsatisfies E. We describe how to extend τ to weight k′ truth assignment τ ′ to the variablesV that satisfies E ′ as follows:(1) For each i such that τ(xi) = 1 and for j = 0, ..., (1 + 2h)22h set τ ′(x[i, j]) = 1.(2) For each i = 1, ...,m choose choose an index ji such that Ei,ji evaluates to 1 (this ispossible, since τ satisfies E) and set τ ′(u[i, ji]) = 1.(3) If in (2) Ei,ji is a sum-of-products, then choose an index ki such that

mi,j,ki∏l=1

x[i, j, k, l]

evaluates to 1, and correspondingly set τ ′(w[i, j, ki]) = 1.(4) For i = 1, ...,m and j = 1, ...,m′i such that j 6= ji, set τ ′(w[i, j, 0]) = 1.

It is straightforward to check that the above described weight k′ extension τ ′ satisfiesE ′.

Conversely, suppose υ′ is a weight k′ truth assignment to the variables of V that satisfiesE ′. We argue that the restriction υ of υ′ to V0 is a weight k truth assignment that satisfiesE.

Claim 1. υ sets at most k variables of V0 to 1.

If this were not so, then the clauses in E1 and E2 would together force at least (k +1)(2 + (1 + 2h)22h) variables to be 1 in order for υ′ to satisfy E ′, a contradiction as this ismore than k′.

Claim 2. υ sets at least k variables of V0 to 1.

The clauses of E4 insure that υ′ sets at most m variables of V2 to 1. The clauses of E5

insure that υ′ sets at most∑mi=1m

′i variables of V3 to 1. If Claim 2 were false then for υ′ to

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have weight k′ there must be more than k indices j for which some variable x[i, j] of V1 hasthe value 1, a contradiction in consideration of E1 and E2.

The clauses of E3 and the arguments above show that υ′ necessarily has the followingrestricited form:(1) Exactly k variables of V0 are set to 1.(2) For each of the k in (1) the corresponding (1 + 2h)22h + 1 variables of V1 are set to 1. (3)For each i = 1, ...,m there is exactly one ji for which u[i, ji] ∈ V2 is set to 1.(4) For each i = 1, ...,m and j = 1, ...,m′i there is exactly one ki for which w[i, j, ki] ∈ V3 isset to 1.

To argue that υ satisfies E it suffices to argue that υ satisfies every Ei,ji for i = 1, ...,m.

The clauses of E6 insure that if υ′(u[i, j]) = 1 then ki 6= 0. This being the case, theclauses of E7 force the literals in a subexpression of Ei,ji to evaluate in a way that showsEi,ji to evaluate to 1. The clauses of E8 enforce that Ei,ji evaluates to 1 for ji > m′i. 2

Thus we have the following stratification of W [1] that will be useful to our arguments.

W [1] =∞⋃s=1

W [1, s]

Our main result shows that W [1] collapses to W [1, 2].

3. Antimonotonicity

A family of circuits F is termed monotone if the circuits in F do not have any notgates. Equivalently, the circuits in F correspond to boolean expressions having only positiveliterals. If we restrict the definition of W [t] and W [1, s] to monotone circuit families weobtain the family of parameterized problems monotone W [t] (monotone W [1, s]).

We say that a family of circuits F is antimonotone if the boolean expressions corre-sponding to the circuits in F have only negative literals. In an antimonotone circuit eachfan-out line from an input node goes to a not gate (and in the remainder of the circuit thereare no other not gates). The restriction to antimonotone circuit families yields the classes ofparameterized problems antimonotone W [t] (antimonotone W [1, s]).

Theorem 3.1 of [DF2] employed as a change-of-variables as in the proof of theorem 4.1of that paper shows the following relationship.

Proposition 3.1. W [t] = monotone W [t] for t even and t ≥ 2. 2

We prove the following complementary result.

Proposition 3.2. W [t] = antimonotone W [t] for t odd, t ≥ 1.

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We first prove the following lemma.

Lemma 3.1 W [1, s] = antimonotone W [1, s] for all s ≥ 2.

Proof. The plan of our argument is to identify a problem (RED/BLUE NONBLOCKER)that belongs to antimonotone W [1, s], and then show that the problem is hard for W [1, s].RED/BLUE NONBLOCKER is the parameterized problem which takes as input a graphG = (V,E) where V is partitioned into two color classes V = Vred ∪ Vblue, and a positiveinteger k. The problem is to determine if there is a set of red vertices V ′ ⊆ Vred of cardinalityk such that every blue vertex has at least one neighbor that does not belong to V ′.

The closed neighborhood of a vertex u ∈ V is the set of vertices N [u] = {x : x ∈V and x = u or xu ∈ E}.

It is easy to see that the restriction of RED/BLUE NONBLOCKER to graphs G ofmaximum degree s belongs to antimonotone W [1, s] since the product-of-sums boolean ex-pression ∏

u∈Vblue

∑xi∈N [u]∩Vred

¬xi

has a weight k truth assignment if and only if G has size k nonblocking set. By the weight ofa truth assignment to a set of boolean variables, we mean the number of variables assignedthe value true.

Such an expression corresponds directly to a circuit meeting the defining conditions forantimonotone W [1, s]. We will refer to the restriction of RED/BLUE NONBLOCKER tographs of maximum degree bounded by s as s-RED/BLUE NONBLOCKER. We next arguethat s-RED/BLUE NONBLOCKER is complete for W [1, s].

Let X be a boolean expression in conjunctive normal form with clauses of size boundedby s. Suppose X consists of m clauses C1, ..., Cm over the set of n variables x0, ..., xn−1. Weshow how to produce in polynomial-time by local replacement, a graph G = (Vred, Vblue, E)that has a nonblocking set of size 2k if and only if X is satisfied by a truth assignment ofweight k.

Before we give any details, we give a brief overview of the construction, whose componentdesign is outlined in Diagram 1. There are 2k ‘red’ components arranged in a circle. Theseare alternatively gouped as blocks from V1 and then V2 sets to be precisely described below.The idea is that V1 blocks should represent a positive choice (corresponding to a literalbeing true) and the V2 blocks corresponding to the ‘gap’ until the next positive choice. Wewill ensure that for each pair in a block there will be a blue vertex connected to the pairand nowhere else (these are the sets V3 and V5). This device ensures that at most one redvertex from each block can be chosen and since we must choose 2k this ensures that wechoose exactly one red vertex from each block. The reader should think of the V2 blocks

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as arranged in k columns. Now if i is chosen fron a V1 block we will ensure that column igets to select the next gap. To ensure this we connect a blue degree 2 vertex to i and eachvertex not in the i-th column of the next V2 block. Of course this means that if i is chosensince these blue vertices must have an unchosen red neighbour, we must choose from thei-th column. The final part of the component design is to enforce consistency in the nextV1 block. That is if we choose i and have a gap choice in the next V2 block, column i, ofj then the next chosen variable should be i + j + 1 (here we work mod n). Again we canenforce this by using many degree 2 blue vertices to block any other choice (These are theV6 vertices.) The last part of the construction is to force consistency with the clauses. Wedo this as follows. For each way a nonblocking set can correspond to making a clause false,we make a blue vertex and join it up to the s relevant vertices. This ensures that thay can’tall be chosen. (This is the point of the V7 vertices.) We now turn to the formal details.

The red vertex set Vred of G is the union of the following sets of vertices:V1 = {a[r1, r2] : 0 ≤ r1 ≤ k − 1, 0 ≤ r2 ≤ n− 1}V2 = {b[r1, r2, r3] : 0 ≤ r1 ≤ k − 1, 0 ≤ r2 ≤ n− 1, 1 ≤ r3 ≤ n− k + 1}

The blue vertex set Vblue of G is the union of the following sets of vertices:V3 = {c[r1, r2, r′2] : 0 ≤ r1 ≤ k − 1, 0 ≤ r2 < r′2 ≤ n− 1}V4 = {d[r1, r2, r

′2, r3, r

′3] : 0 ≤ r1 ≤ k−1, 0 ≤ r2, r

′2 ≤ n−1, 0 ≤ r3, r

′3 ≤ n−1 and either r2 6=

r′2 or r3 6= r′3}V5 = {e[r1, r2, r′2, r3] : 0 ≤ r1 ≤ k − 1, 0 ≤ r2, r

′2 ≤ n− 1, r2 6= r′2, 1 ≤ r3 ≤ n− k + 1}

V6 = {f [r1, r′1, r2, r3] : 0 ≤ r1, r

′1 ≤ k−1, 0 ≤ r2 ≤ n−1, 1 ≤ r3 ≤ n−k+1, r′1 6= r2+r3 mod n}

V7 = {g[j, j′] : 1 ≤ j ≤ m, 1 ≤ j′ ≤ mj}

In the desription of V7, the integers mj are bounded by a polynomial in n and k ofdegree a function of s which will be described below. Note that since s is a fixed constantindependent of k, this is allowed by our definition of reduction for parameterized problems.

For convenience we distinguish the following sets of vertices.A(r1) = {a[r1, r2] : 0 ≤ r2 ≤ n− 1}B(r1) = {b[r1, r2, r3] : 0 ≤ r2 ≤ n− 1, 1 ≤ r3 ≤ n− k + 1}B(r1, r2) = {b[r1, r2, r3] : 1 ≤ r3 ≤ n− k + 1}

The edge set E of G is the union of the following sets of edges. In these descriptions weimplicitly quantify over all possible indices for the vertex sets V1, ..., V7.E1 = {a[r1, q]c[r1, r2, r

′2] : q = r2 or q = r′2}

E2 = {b[r1, q2, q3]d[r1, r2, r′2, r3, r

′3] : either (q2 = r2 and q3 = r3) or (q2 = r′2 and q3 = r′3)}

E3 = {a[r1, r2]e[r1, r2, q, q′]}

E4 = {b[r1, q, q′]e[r1, r2, q, q′]}E5 = {b[r1, r2, r3]f [r1, r

′1, r2, r3]}

E6 = {a[r1 + 1 mod n, r′1]f [r1, r′1, r2, r3]}

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We say that a red vertex a[r1, r2] represents the possibility that the boolean variablexr2 may evaluate to true (corresponding to the possibility that a[r1, r2] may belong to a 2k-element nonblocking set V ′ in G). Similarly, we say that a red vertex b[r1, r2, r3] representsthe possibility that the boolean variables xr2+1, ..., xr2+r3−1 (with indices reduced mod n) mayevaluate to false.

Suppose C is a clause of X having s literals. There are O(n2s) distinct ways of choosing,for each literal l ∈ C, a single vertex representative of the possibility that l = xi may evaluateto false, in the case that l is a positive literal, or in the case that l is a negative literal l = ¬xi,a representative of the possibility that xi may evaluate to true. For each clause Cj of X,j = 1, ...,m, let R(j, 1), R(j, 2), ..., R(j,mj) be an enumeration of the distinct possibilities forsuch a set of representatives. We have the additional sets of edges for the clause componentsof G:E7 = {a[r1, r2]g[j, j′] : a[r1, r2] ∈ R(j, j′)}E8 = {b[r1, r2, r3]g[j, j′] : b[r1, r2, r3] ∈ R(j, j′)}

Suppose X has a satisfying truth assigment τ of weight k, with variables xi0 , xi1 , ..., xik−1

assigned the value true. Suppose i0 < i2 < ... < ik−1. Let dr = ir+1(modk) − ir (mod n) forr = 0, ..., k − 1. It is straightforward to verify that the set of 2k vertices

N = {a[r, ir] : 0 ≤ r ≤ k − 1} ∪ {b[r, ir, dr] : 0 ≤ r ≤ k − 1}

is a nonblocking set in G.

Conversely, suppose N is a 2k-element nonblocking set in G. It is straightforward tocheck that a truth assignment for X of weight k is described by setting those variables truefor which a vertex representative of this possibility belongs to N , and by setting all othervariables to false.

Note that the edges of the sets E1 (E2) which connect pairs of distinct vertices of A(r1)(B(r1)) to blue vertices of degree two, enforce that any 2k-element nonblocking set mustcontain exactly one vertex from each of the sets A(0), B(0), A(1), B(1), ..., A(k−1), B(k−1).The edges of E3 and E4 enforce (again by connections to blue vertices of degree two) thatif a representative of the possibility that xi evaluates to true is selected for a nonblockingset from A(r1), then a vertex in the ith row of B(r1) must be selected as well, representing(consistently) the interval of variables set false (by increasing index modn) until the “next”variable selected to be true. The edges of E5 and E6 insure consistency between the selectionin A(r1) and the selection in A(r1 + 1 mod n). The edges of E7 and E8 insure that aconsistent selection can be nonblocking if and only if it does not happen that there is a setof representatives for a clause witnessing that every literal in the clause evaluates to false.(There is a blue vertex for every such possible set of representatives.) 2

Proof of Prop. 3.2. Let C be a circuit of weft t for t odd, t ≥ 3. By Theorem 4.1 of [DF2]we may assume that C is represented by a boolean expression E0 that is in (alternating)

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product-of-sums-of-products... form (for t alternations). The first level of the circuit belowthe inputs consists of And gates (since t is odd).

Suppose the inputs to C are x1, ..., xn. Let X1 be the boolean expression with single-literal clauses X1 = (x1)(x2) · · · (xn) and let G be the graph constructed from X1 by thereduction in the lemma above. Let y1, ..., yz be new variables, one for each red vertex in G.

Let E1 be the boolean expression

E1 =∏

u∈(Vblue−V7)

∑yi∈N [u]

¬yi

and let C1 be a circuit realizing E1.

We modify C in the following ways:(1) Each positive fan-out from an input xi to C is replaced by an And gate receiving negatedinputs from all of the new input variables yj for which the corresponding red vertices of Grepresent the possibility that xi evaluates to false.(2) Each negated fan-out from an input xi to C is replaced by an And gate receiving negatedinputs from all of the new input variables yj for which the corresponding red vertices of Grepresent the possibility that xi evaluates to true.(3) The circuit C1 is conjunctively combined with C at the bottommost (output) And gate.

The circuit C ′ obtained in this way accepts a weight 2k input vector if and only if Caccepts a weight k input vector. The argument for correctness is essentially the same as forLemma 3.1. The circuit C ′ has weft t after the And gates replacing the former inputs arecoalesced with the And gates of the topmost large gate level (this is feasible, since t is odd).All of the input fan-out lines of C ′ are negated. 2

Lemma 3.1 provides the starting point for demonstrating the following collapse of theW [1] stratification.

Proposition 3.3. W [1] = W [1, 2]

Proof. It suffices to argue that for all s ≥ 2, antimonotone W [1, s] = W [1, 2]. The ar-gument here consists of another change of variables. Let C be an antimonotone W [1, s]circuit for which we wish to determine whether a weight k input vector is accepted. Weshow how to produce a circuit C ′ corresponding to an expression in conjunctive normal formwith clause size two, that accepts an input vector of weight

k′ = k2k +s∑i=2

(k

i

)if and only if C accepts an input vector of weight k. (The circuit C ′ will in general notbe antimonotone, but this is immaterial by Lemma 3.1. Actually in [DEF] we use another

reduction that only needs k′ = ks+1 +∑si=2

(ki

)and is hence polynomial in k for a fixed s.)

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Let x[j] for j = 1, ...n be the input variables to C. The idea is to create new variablesrepresenting all possible sets of at most s and at least 2 of the variables x[j]. Let A1, ..., Apbe an enumeration of all such subsets of the input variables x[j] to C. The inputs to eachor gate g in C (all negated, since C is antimonotone) are precisely the elements of some Ai.The new input corresponding to Ai represents that all of the variables whose negations areinputs to the gate g have the value true. Thus in the construction of C ′, the or gate g isreplaced by the negation of the corresponding new “collective” input variable.

We introduce new input variables of the following kinds:(1) One new input variable v[i] for each set Ai for i = 1, ..., p, to be used as above.(2) For each x[j] we introduce 2k copies x[j, 0], x[j, 1], x[j, 2], ..., x[j, 2k − 1].

In addition to replacing the or gates of C as described above, we add to the circuitadditional or gates of fan-in 2 that provide an enforcement mechanism for the change ofvariables. The necessary requirements can be easily expressed in conjunctive normal formwith clause size two, and thus can be incorporated into a W [1, 2] circuit.

We require the following implications concerning the new variables:

(1) The n · 2k implications, for j = 1, ..., n and r = 0, ..., 2k − 1,

x[j, r]⇒ x[j, r + 1 (mod 2k)]

(2) For each containment Ai ⊆ Ai′ , the implication

v[i′]⇒ v[i]

(3) For each membership x[j] ∈ Ai, the implication

v[i]⇒ x[j, 0]

It may be seen that this transformation may increase the size of the circuit by a lin-ear factor exponential in k. We make the following argument for the correctness of thetransformation.

If C accepts a weight k input vector, then setting the corresponding copies x[i, j] amongthe new input variables accordingly, together with appropriate settings for the the new“collective” variables v[i] yields a vector of weight k′ that is accepted by C ′.

For the other direction, suppose C ′ accepts a vector of weight k′. Because of the impli-cations in (1) above, exactly k sets of copies of inputs to C must have value 1 in the acceptedinput vector (since there are 2k copies in each set). Because of the implications described in(2) and (3) above, the variables v[i] must have values in the accepted input vector compatiblewith the values of the sets of copies. By the construction of C ′, this implies there is a weightk input vector accepted by C. 2

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We have now done most of the work required to show that the following well- knownproblems are complete for W [1].

INDEPENDENT SETInstance: A graph G = (V,E) and a positive integer k.Question: Is there a set V ′ ⊆ V of cardinality k, such that ∀u, v ∈ V ′, uv /∈ E?

CLIQUEInstance: A graph G = (V,E) and a positive integer k.Question: Is there a set of k vertices V ′ ⊆ V that forms a complete subgraph of G (that is,a clique of size k)?

Theorem 3.1. INDEPENDENT SET is complete for W [1].

Proof. It is easy to observe that INDEPENDENT SET belongs to W [1]. By Lemma 3.1and Theorem 3.1 it is enough to argue that INDEPENDENT SET is hard for antimonotoneW [1, 2]. Given a boolean expression X in conjuctive normal form (product of sums) withclause size two and all literals negated, we may form a graph GX with one vertex for eachvariable of X, and having an edge between each pair of vertices corresponding to variablesin a clause. The graph GX has an independent set of size k if and only if X has a weight ktruth assignment. 2

Corollary 3.2 CLIQUE is complete for W [1].

Proof. This follows immediately by considering the complement of a given graph. Thecomplement has an independent set of size k if and only if the graph has a clique of size k.2

4. Problems Hard for W [1]

In this section we show that the following problems are hard for W [1]. None of themis presently known to belong to W [1]. We conjecture that the first two problems, which areshown to be equivalent with respect to uniform reductions, and to belong to W [2], are ofdifficulty intermediate between W [1] and W [2].

PERFECT CODEInstance: A graph G = (V,E) and a positive integer k.Question: Does G have a k-element perfect code? A perfect code is a set of vertices V ′ ⊆ Vwith the property that for each vertex v ∈ V there is precisely one vertex in N [v] ∩ V ′.

WEIGHTED EXACT CNF SATISFIABILITYInstance: A boolean expression E in conjuctive normal form, and a positive integer k.Question: Is there a truth assignment of weight k to the variables of E that makes exactly

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one literal in each clause of E true?

SIZED SUBSET SUMInstance: A list of positive integers L = (x1, x2, ..., xn), a positive integer S and a positiveinteger k.Question: Is there a sublist of L of size k that sums to S?

Lemma 4.1. PERFECT CODE ∈ W [2].

Proof. Let G = (V,E) be a graph for which we wish to determine whether G has a k-element perfect code. It suffices to show how to efficiently construct a boolean expressionEG in product-of-sums form that has a weight k truth assignment if and only if the graphG has a k-element perfect code. Let EG be the expression E = E0E1E2 where the variablesof EG are in one-to-one correspondence with vertices of G and

E0 =∏u∈V

∑x∈N [u]

E1 =∏uv∈E

(¬u+ ¬v)

E2 =∏

uv,vw∈E(¬u+ ¬w)

If G has a k-element perfect code V ′ ⊆ V , then the truth assignment which sets the variablescorresponding to the vertices of V ′ true and all others false satisfies EG, since V ′ is anindependent set (so that E1 is satisfied), and V ′ contains no vertices at a distance 2 fromeach other in G (so that E2 is satisfied), and yet V ′ is dominating set (so that E2 is satisfied).Conversely, any satisfying truth assignment for EG of weight k must satisfy each of of thesesubproducts, and therefore the vertices corresponding to the variables set to true must be aperfect code in G. 2

Lemma 4.2. PERFECT CODE reduces to WEIGHTED EXACT CNF SATISFIABILITY.

Proof. A graph G has a k-element perfect code if and only if the expression E0 constructedas in Lemma 5.1 has a weight k truth assignment that makes exactly one literal in eachclause true. 2

Lemma 4.3. WEIGHTED EXACT CNF SATISFIABILITY reduces to PERFECT CODE.

Proof. The reduction can be demonstrated using the transformation used in the proof of The-orem 3.1 of [DF1] (which is there used to reduce Weighted Satisfiability to DOMINATINGSET). 2

Lemma 4.4. PERFECT CODE reduces to SIZED SUBSET SUM.

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Proof. Let G = (V,E) be a graph for which we wish to determine whether G has a perfectcode of size k. Suppose for convenience that the vertex set of the graph V = {0, ..., n− 1}.We can easily compute the list of positive integers L = (x[i, j] : 1 ≤ i ≤ k, 0 ≤ j ≤ n − 1)and the positive integer M , where

x[i, j] = (k + 1)n+k−i +∑

u∈N [j]

(k + 1)u

M =n+k−1∑t=0

(k + 1)t

such that L has a sublist of size k summing to M if and only if G has a k-element perfectcode. The correctness of this transformation is easily observed if the numbers of L arerepresented in base k+ 1, and it is noted that there can be no carries in a sum of k integersfrom L expressed in this way. 2

Theorem 4.1. PERFECT CODE is hard for W [1].

Proof. We reduce from INDEPENDENT SET. Let G = (V,E) be a graph. We show

how to produce a graph H = (V ′, E ′) that has a perfect code of size k′ =(k2

)+ k + 1 if and

only if G has a k-element independent set. The vertex set V ′ of H is the union of the setsof vertices:V1 = {a[s] : 0 ≤ s ≤ 2}V2 = {b[i] : 1 ≤ i ≤ k}V3 = {c[i] : 1 ≤ i ≤ k}V4 = {d[i, u] : 1 ≤ i ≤ k, u ∈ V }V5 = {e[i, j, u] : 1 ≤ i < j ≤ k, u ∈ V }V6 = {f [i, j, u, v] : 1 ≤ i < j ≤ k, u, v ∈ V }

The edge set E ′ of H is the union of the sets of edges:E1 = {a[0]a[i] : i = 1, 2}E2 = {a[0]b[i] : 1 ≤ i ≤ k}E3 = {b[i]c[i] : 1 ≤ i ≤ k}E4 = {c[i]d[i, u] : 1 ≤ i ≤ k, u ∈ V }E5 = {d[i, u]d[i, v] : 1 ≤ i ≤ k, u, v ∈ V }E6 = {d[i, u]e[i, j, u] : 1 ≤ i < j ≤ k, u ∈ V }E7 = {d[j, v]e[i, j, u] : 1 ≤ i < j ≤ k, v ∈ N [u]}E8 = {e[i, j, x]f [i, j, u, v] : 1 ≤ i < j ≤ k, x 6= u, x /∈ N [v]}E9 = {f [i, j, u, v]f [i, j, x, y] : 1 ≤ i < j ≤ k, u 6= x or v 6= y}

An overview of this construction is given in Diagram 2 and a (partial) example is givenin Diagram 3. Suppose C is a perfect code of size k′ in H. Since a[1] and a[2] are pendantvertices attached to a[0], neither vertex belongs to C because both cannot belong to C, andif only one belongs to C, then C fails to be a dominating set. It follows that a[0] ∈ C. This

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implies that none of the vertices in V2 and V3 belong to C (V3 would kill V2), and it impliesalso that exactly one vertex in each of the cliques formed by the edges of E5 belongs to C(to cover V3). Note that each of these k cliques has n vertices indexed by V , the vertex setof G (this is the selection gadget). Let I be the set of vertices of G corresponding to theelements of C in these cliques. We argue that I is an independent set of order k in G.

Suppose u, v ∈ I and that uv ∈ E. Then there are indices i < j between 1 and k suchthat (without losss of generality) d[i, u] ∈ C and d[j, v] ∈ C. By the definition of E6 and E7

each of these vertices is adjacent to e[i, j, u], which contradicts that C is a perfect code inH. Thus I is an independent set in G.

Conversely, we argue that if J = {u1, ..., uk} is a k-element independent set in G, thenH has a perfect code CJ of size k′. We may take CJ to be the following set of vertices:

CJ = {a[0]} ∪ {d[i, ui] : 1 ≤ i ≤ k} ∪ {f [i, j, ui, uj] : 1 ≤ i, j ≤ k}

That CJ is a perfect code can be verified directly from the definition of H. 2

By Lemmas 4.2, 4.4 and the above theorem we have the following hardness results aswell.

Theorem 4.2 WEIGHTED EXACT CNF SATISFIABILITY is hard for W [1]. 2

Theorem 4.3 SIZED SUBSET SUM is hard for W [1]. 2

One problem that we are quite interested in is the natural analogue of Travelling Sales-person:

SHORT CHEAP TOURInstance: A weighted graph and positive integers S and k.Question: Is there a tour through at least k vertices of cost at most S?

The precise difficulty of this problem is at present open but a variation is hard for W [1].Let SHORT EXACT TOUR be the same as SHORT CHEAP TOUR except that we askthat the tour costs exactly S.

Theorem 4.4 SHORT EXACT TOUR is hard for W [1].

Proof. Let (L, S, k) be an instance of sized subset sum,with L = {x1, ..., xn}. Construct agraph G as follows: For each xi we have two vertices yi and zi . Join yi to zi with an edge ofweight xi. Let d exceed x1 + ... + xn. For i not equal to j join yi to zj. Give all such edgesweight d. Now ask if G has a 2k vertex tour of weight S + kd? 2

The reader should note that the natural analogue of HAMILTON CIRCUIT which asks

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if there is a cycle through k or more vertices is strongly uniformly fixed parameter tractable(Bodlaender), but it is unknown if the problem of determining if there is a cycle of sizeexactly k is also tractable. (See [JvL,section 2.4.3]).

As a final example, we remark that the reduction of [DF2] can be used to show that thefollowing problem is also hard for W [1].

WEIGHTED EXACT BINARY INTEGER PROGRAMMINGInstance: A binary vector b, a binary matrix A and an integer k.Question: Is there a binary vector x of weight k such that Ax equals b?

5. Open Problems

The study of fixed-parameter tractability and completeness can be viewed as addressingaspects of the the general subject of computational infeasibility inside of P . For relatedwork examining limited ammounts of nondeterminism see [BG]. Many familiar issues incomplexity theory have unexplored analogues in the fixed-parameter setting (such as paralleland randomized complexity, one-way functions, and approximation). A number of basicstructural questions concerning the W hierarchy have yet to be resolved. For example,while it is known a collapse of the W hierarchy implies a collapse involving more familiarunparameterized complexity classes ([ADF2]), the exact relationship is unknown.

A wide variety of natural parameterized problems may well be complete for variouslevels of the W hierarchy. Well-known natural problems for which neither fixed-parametertractability nor W [t] hardness is presently known include: DIRECTED FEEDBACK VER-TEX SET, GRAPH TOPLOGICAL CONTAINMENT and IMMERSION ORDERING (theparameters in the last two problems being a fixed Graph.) (for the definitions, see [GJ]).

6. Addendum 7 Feb 1994

Since the original writing of this paper, there has benn quite a bit of activity regardingW [1] and it is clear that this is probably the most important class one can use to establishfixed parameter intractability along the lines of establishing intractability via NP com-pleteness. Particularly strong evidence for the intractability of W [1] is given in Cai, Chen,Downey, and Fellows [CCDF] where it is established that the following very generic problemis W [1] complete:

SHORT TURING MACHINE COMPUTATIONInput: A Nondeterministic Turing Machine M and a string x.Parameter: k.Question: Does M have a length k computation path accepting x?

This problem is particularly signifigant as it proves a sort of Cook’s theorem in a pa-

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rameterized setting. Many other problems have been shown to be W [1] complete. We quotea couple. In [CCDF] it is also proven that the following are W [1] complete:

SHORT DERIVATION (for unrestricted grammars)Input: A phrase-structure grammar G and a word x.Parameter: kQuestion: Is there a G derivation of x of length k?

SHORT POST CORRESPONDENCEInput: A Post Correspondence System Π.Parameter: kQuestion: Is there a length k solution for Π?

Downey, Fellows, Kapron, Hallett, and Wareham [DFKHW] proved that the followingproblem is W [1] complete:

SHORT CSL DERIVATIONInput: A context sensitive grammar G and a word x ∈ Σ∗.Parameter: kQuestion: Is there a G derivation of x of length at most k?

Downey and Fellows proved that some parameterized versions of embedding questionsturn out to be W [1] complete. For instance from [DF7] we have the following being W [1]complete.

SEMIGROUP EMBEDDINGInput: A semigroup G.Parameter: A semigroup H.Question: Is H embeddable into G?

SEMILATTICE EMBEDDINGInput: A semialttice S.Parameter: A semialttice L.Question: Is L embeddable into S?

BIPARTITE GRAPH EMBEDDINGInput: A bipartite graph G.Parameter: A bipatrite graph H.Question: Is H embeddable into G?

Another area that has found W [1] complete problems is that of Computational LearningTheory. Consider the following problem which is the most important parameter in learningtheory.

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VAPNIK CHERVONENKIS DIMENSIONInput: A family F of subsets of a base set X.Parameter: k.Question: Is the VC-dimension of F at least k?

In [DEF], the authors together with P. Evans proved that VAPNIK-CHERVONENKISDIMENSION is hard for W [1], and hence combined with membership of W [1] which is provenin Downey-Fellows [DF5], we see that this problem is W [1] complete. We remark that thisis very interesting since the unparameterized version is highly unlikely to be NP -completeunless NP is very small. See, Papadimitriou and Yannakakis [PY].

Finally we mention some problems that are W [1] complete arizing from molecular biol-ogy, which is a particularly fertile area of applications for this theory in view of the fact thatmany problems have small parameters (such as the number of strands of DNA) yet largeproblem size.

LONGEST COMMON SUBSEQUENCEInput: A set of k strings X1, ..., Xk over Σ∗.Parameter[1]: k.Parameter[2]: m.Parameter[3]: m, k.Question: Is there a string X ∈ Σ∗ that has at least m symbols that is a subsequence ofX1, ..., Xk?

In Bodlaender, Downey, Fellows, and Wareham [BDFW] it is shown that all of the vari-ations LCS[i] (with the obvious meanings) are W [1] hard, and that LCS[3] is W [1] complete.

We conclude by remarking that there have beeen very many other problems which havebeen proven to be W [1] hard and even complete for other levels of the W -hierarchy. Partiallists can be found in [DF2] and [DF4].

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References

[ADF1] K. A. Abrahamson, R.G. Downey and M. F. Fellows,“ Fixed-Parameter IntractabilityII,” In STACS’93 (1993) Springer-Verlag Lecture Notes in Computer Science, 374-385.

[ADF2] K. A. Abrahamson, R. G. Downey, and M. R. Fellows, “Fixed-Parameter Tractabilityand Completeness IV: On Completeness for W [P ] and PSPACE Analogues”, to appear,Annals of Pure and Applied Logic.

[BG] J. F. Buss and J. Goldsmith, “Nondeterminism Within P ,” SIAM J of Comput. 22(1993) 560-572.

[Bo] H. L. Bodlaender, “On Disjoint Cycles,” Technical Report RUU-CS-90-29, Dept. ofComputer Science, Utrecht University, Utrecht, The Netherlands, August 1990.

[BDFW] H. L. Bodlaender, R. G. Downey, M. R. Fellows, and H. Todd Wareham,“TheParameterized Complexity of Sequence Alignment and Concensus,” In Proceedings Combi-natorial Pattern Matching, Asilomar, 1994, to appear.

[CCDF] L. Cai, J. Chen, R. G. Downey and M. F. Fellows,“The Parameterized Complexityof Short Computation and Factorization,” submitted.

[DEF] R. G. Downey, P. Evans, and M. F. Fellows,“Parameterized Learning Complexity,”inCOLT’93 (6 th Annual Conference on Learning Theory, 1993) IEEE publ. 51-57.

[DF1] R. G. Downey and M. R. Fellows, “Fixed-Parameter Tractability and Completeness,Cong. Numer. 87 (1992) 161-187.

[DF2] R. G. Downey and M. R. Fellows, “Fixed-Parameter Tractability and CompletenessI: Basic Results,” to appear SIAM J. of Comput.

[DF3] R. G. Downey and M. R. Fellows, “Fixed-Parameter Tractability and CompletenessIII: Some Structural Aspects of the W -Hierarchy,” in Complexity Theory, (ed. K. Ambos-Spies, S. Homer, and U. Sch”/oing), Cambridge University Press, (1993) 166-191.

[DF4] R. G. Downey and M. F. Fellows,“Fixed-Parameter Intractability,” Proceedings 7thAnnual Structure in Complexity (1992) 36-49.

[DF5] R. G. Downey and M. F. Fellows,“Parameterized Computational Feasibility,”in Fea-sible Mathematics II, (ed. P. Clote and J. Remmel) Birkhauser, Boston, to appear.

[DF6] R. G. Downey and M. F. Fellows,Parameterized Complexity Monograph in preparation.

[DF7] R. G. Downey and M. F. Fellows,“Fixed-Parameter Tractability and Intractability-a

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Survey,” in preparation.

[FL1] M. R. Fellows and M. A. Langston, “On Search, Decision and the Efficiency ofPolynomial-Time Algorithms.” In Proc. Symp. on Theory of Computing (STOC) (1989),501-512.

[FL2] M. R. Fellows and M. A. Langston, “An Analogue of the Myhill-Nerode Theoremand Its Use in Computing Finite Basis Characterizations.” In Proc. Symp. Foundations ofComp. Sci. (FOCS) (1989), 520-525.

[GJ] M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theoryof NP -Completeness (Freeman, San Francisco, 1979).

[PY] C. Papadimitriou and M. Yannakakis,“On limited Nondeterminism and the Complexityof the VC-Dimension,”in Proceedings 8th Annual Structure in Complexity Conference, IEEEPubl. (1993) 12-18.

[RS1] N. Robertson and P. D. Seymour, “Graph Minors XIII. The Disjoint Paths Problem,”to appear.

[RS2] N. Robertson and P. D. Seymour, “Graph Minors XV. Wagner’s Conjecture,” toappear.

[JvL] J. van Leeuwen, “Graph Algorithms.” in Handbook of Theoretical Computer Science,Volume A (ed J. van Leeuwen) Elsevier Science Publishers B.V.,The Netherlands (1990),525-632.

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